Efficient resource state distillation

ABSTRACT

Systems and methods are provided for generating at least one high fidelity resource state. A classical code and punctured to provide a first set of generators and a second set of generators. The first set of generators is mapped to a set of stabilizer generators, and the second set of generators is mapped to a set of logical operators. A set of resource states are prepared in physical qubits. A decoding process is performed on the resource states according to a quantum code represented by the set of stabilizer generators and the set of logical operators, and qubits corresponding to the stabilizers are measured.

RELATED APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 13/765,332, filed 12 Feb. 2013, which claims priority from U.S.Patent Application Ser. No. 61/719,063, filed 26 Oct. 2012, both ofwhich are incorporated herein in their entirety.

TECHNICAL FIELD

This invention relates to quantum computing, and more particularly, to amethod of efficient resource state distillation.

BACKGROUND

A classical computer operates by processing bits (binary digits) ofinformation that obey the laws of classical physics. The bits arephysically represented by a high or a low energy level, correspondingeither to a logical one (e.g., high energy) or a logical zero (e.g., lowenergy). Transformations on these information bits can be describedusing simple classical logic gates such as AND and OR gates. A classicalalgorithm, such as one that multiplies two integers, can be decomposedinto a long string of these simple logic gates. Like a classicalcomputer, a quantum computer also has bits and gates. Instead of storingexclusively logical ones and zeroes, a quantum bit (“qubit”) can storeany quantum mechanical superposition of the two, in some sense allowinga qubit to be in both classical states simultaneously. This ability,coupled with quantum gates that allow the production of suchsuperpositions, enables a quantum computer to solve certain problemswith exponentially greater efficiency than that of a classical computer.

SUMMARY OF THE INVENTION

In accordance with an aspect of the present invention, a method isprovided for generating at least one high fidelity resource state, wherethe states are a resource for implementing rotations by some angle π/g.A classical code is selected in which all of the codewords associatedwith the code have a weight of zero mod 2g, where g is a positiveinteger power of two. The classical code is punctured such that a firstset of codewords retain a weight of zero mod 2g and a second set ofcodewords has a weight of 2g−1 mod 2g. The first set of codewords ismapped to a set of stabilizer generators, and the second set ofcodewords is mapped to a set of logical operators. A set of relativelylow fidelity resource states are prepared in a plurality of physicalqubits. A decoding process is performed on the set of relatively lowfidelity resource states according to a quantum code represented by theset of stabilizer generators and the set of logical operators. Each ofthe physical qubits corresponding (after decoding) to Z stabilizers aremeasured and the result is used to determine a correction to be made tothe other qubits prior to measuring those corresponding to Xstabilizers.

In accordance with another aspect of the present invention, a system isprovided for generating at least one high fidelity resource state from aplurality of relatively low fidelity resource states. An encodingcircuit encodes a set of logical qubits into a plurality of physicalqubits according to a punctured classical code. A rotation componentconsumes the relatively low fidelity resource state σ to apply arotation to a state stored in each physical qubit around an axis of theBloch sphere of π/g, where g is a positive integer power of two,resulting in a rotation being applied to the logical qubits. A decodingcircuit decodes the rotated logical states stored in the physical qubitsto provide the set of logical qubits stored in a proper subset of theplurality of physical qubits, at least one of the set of logical qubitscomprising a high fidelity resource state.

In accordance with yet another aspect of the present invention, a systemis provided for generating at least one high fidelity magic state for aset of Clifford gates. A plurality of physical qubits each store arelatively low fidelity state approximating |eiπ/4

. A decoder includes a first plurality of quantum gates that implement aunitary decoder on the physical qubits corresponding to a quantum codederived from a punctured classical code having a first set of codewordswith a weight of zero mod 8 and a second set of codewords with a weightof 7 mod 8. The decoder further includes a first plurality ofmeasurement assemblies for measuring a plurality of physical qubits. Acorrection component includes a second plurality of quantum gates, eachconfigured to conditionally apply an associated operation to a statestored in one of the unmeasured physical qubits according to the firstset of measurement values, and a second set of measurement assemblies tomeasure at least one of the unmeasured qubits.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates one example of a system for generating at least onehigh fidelity resource state from a plurality of relatively low fidelityresource states in accordance with an aspect of the present invention;

FIG. 2 illustrates one implementation of the system of FIG. 1 as aquantum circuit for performing magic state distillation in accordancewith an aspect of the present invention;

FIG. 3 illustrates one circuit that might be used in the rotationcomponent of FIG. 2 to perform a π/g Z rotation using a prepared |eiπ/g

ancilla state;

FIG. 4 illustrates another example of a system for generating at leastone high fidelity magic state for the set of Clifford gates from aplurality of relatively low fidelity magic states in accordance with anaspect of the present invention;

FIG. 5 illustrates one implementation of the system of FIG. 4 as aquantum circuit for performing magic state distillation in accordancewith an aspect of the present invention; and

FIG. 6 illustrates a first method for generating at least one highfidelity resource state.

FIG. 7 illustrates a second method for generating at least one highfidelity resource state.

DETAILED DESCRIPTION

In accordance with an aspect of the present invention, a family ofroutines is provided to provide distillation of high fidelity resourcestates from a plurality of relatively low fidelity resource states.Specifically, the systems and methods described herein generatequadratic error suppression with increased efficiency, achieving a stateoverhead, that is, a ratio of the number of low fidelity input states tothe number of improved fidelity output states, that approaches three asthe distillation is scaled to larger numbers of input low fidelitystates.

A standard basis for qubit states includes a zero state, denoted hereinas |0> and a one state, orthogonal to the zero state and denoted hereinas |1>. A sign basis includes a plus state, |+>, defined in the standardbasis as 1/√{square root over (2)}|0

+1/√{square root over (2)}|1

, and a minus state, |−>, defined in the standard basis as 1/√{squareroot over (2)}|0

−1/√{square root over (2)}|1

. A Y basis includes +1 state, |Y₊>, defined in the standard basis as1/√{square root over (2)}|0

+i/√{square root over (2)}|1

, and a −1 state, |Y⁻>, defined in the standard basis as 1/√{square rootover (2)}|0

−i/√{square root over (2)}|1

.

A set of universal quantum gates, as the phrase is used herein, is anyset of quantum gates to which any operation possible on a quantumcomputer can be reduced, that is, any other unitary operation can beapproximated (to some arbitrary accuracy) as a finite sequence of gatesfrom the set. Conversely, a non-universal set of gates refers to a setof quantum gates that lacks this property. One example of anon-universal set of gates is the set of Clifford gates. The Cliffordset of gates, as used herein, is defined to include gates implementingeach of the Pauli operators, a controlled not (CNOT) gate, the Hadamardgate, and the π/2 Z rotation gate, as well as a high-fidelityinitialization to the ground state of the standard basis, andmeasurement assemblies for measuring qubit states in the standard basis,the sign basis, and the Y basis.

A “resource state”, as used herein, is a quantum state which can be usedwith a given set of gates to provide a quantum operation that is notincluded in that set or even a quantum operation that is not convenientto generate using that set. It will be appreciated that what constitutesa resource state will vary with the given set of gates. A “magic state”,as used herein, is a quantum state which can be used with the Cliffordset of gates to provide arbitrary quantum operations and can be improvedin fidelity by a circuit constructed using gates from the set ofClifford gates. Effectively, the use of a given magic state, or rather,a finite number of instances of the magic state, with the Clifford gatesallows for universal quantum computation.

“Distillation”, as used herein, refers to the process ofprobabilistically converting a set of qubits in a given state to producea smaller set of qubits that assume the given state with higherfidelity. In quantum information theory, fidelity is a measure of the“closeness” of two quantum states. As used herein, fidelity refers tothe closeness of a given instance of a state to a desired ideal state. A“low fidelity state” refers to a state that is below the level offidelity desirable for use in a quantum computation but above a minimumfidelity needed for a distillation process. In accordance with an aspectof the present invention, resource states can be improved to anarbitrary level of fidelity via repeated applications of statedistillation.

Classical error correcting codes use a syndrome measurement to diagnosewhich error corrupts an encoded state. Quantum error correction alsoemploys syndrome measurements by performing a multi-qubit measurementthat does not disturb the quantum information in the encoded state butretrieves information about the error. Syndrome measurements candetermine whether a qubit has been corrupted, and if so, which one. Infact, the outcome of this operation can indicate not only which physicalqubit was affected, but also, in which of several possible ways it wasaffected. In most codes, errors are projected into either bit flips, orsign (of the phase) flips, or both (corresponding to the Pauli matricesX, Z, and Y). An encoding is used to spread the information associatedwith a group of qubits, referred to as logical qubits, onto a largergroup of physical qubits such that the physical qubits are in aneigenstate of a collection of multi-qubit Pauli operators is referred toherein as a quantum code. A CSS quantum code is a quantum code wherein abasis for the multi-qubit Pauli operators defining the code can beexpressed using multi-qubit Pauli operators each of which are composedof only the single qubit Pauli operators X and I or the single qubitPauli operators Z and I.

FIG. 1 illustrates one example of a system 10 for generating at leastone high fidelity resource state from a plurality of relatively lowfidelity resource states 12 in accordance with an aspect of the presentinvention. The system 10 utilizes a CSS quantum code to encode a set oflogical qubits onto a larger set of physical qubits 14-17. The code isselected to have the property that an encoded −π/g rotation on each ofthe logical qubits can be implemented via transversal application of π/grotations to each of the physical qubits, where g is a positive integerpower of two. To this end, each of the physical qubits 14-17 is preparedwith a state that can be prepared with high-fidelity via a set of gatesnot including π/g rotations.

An encoding circuit 22 encodes the set of logical qubits into aplurality of the physical qubits 14-17 according to a puncturedclassical code having the desired properties. It will be appreciatedthat a first set of the classical code words can represent stabilizersof the quantum code and a second set of the classical code words canrepresent the logical operators. Specifically, the inventor has foundthat a classical code in which each of the code words has a weight ofzero mod 2g is suitable for use in applying a transversal rotation of−π/g. Such a code can be selected and appropriately punctured to providea desired degree of error suppression (e.g., quadratic, cubic, quartic,etc.).

A rotation component 24 applies a rotation to each of the states storedin the physical qubits 14-17 around an axis of the Bloch sphere.Specifically, a rotation of π/g is provided. It will be appreciated thatthe rotation element 24 will consume the plurality of low fidelityresource states to provide the transverse rotation. In oneimplementation, the rotation is made around the Z-axis of the Blochsphere. One example of a quantum circuit that might be used in therotation component is provided in FIG. 3 below.

A decoder 26 decodes the states stored in the physical qubits 14-17 toprovide the set of logical qubits. As part of the decoding process, eachof the physical qubits representing (after unitary decoding) one of thestabilizers is measured in a measurement basis. If the measured valuesassociated with the stabilizers are as desired, the remaining qubits areretained as resource states having a fidelity greater than that of thestates consumed at the rotation component 24. If one or more of thesecond set of measurement values do not assume the correct value, theremaining qubits can be corrected or discarded.

FIG. 2 illustrates one implementation of the system of FIG. 1 as aquantum circuit 30 for performing magic state distillation in accordancewith an aspect of the present invention. In the illustratedimplementation, the magic state is one of a family of states equivalentto the +1 eigenstate of the Hadamard operator. By “equivalent,” it ismeant that each state can be derived from the others via one-qubitClifford unitary operators. The specific magic state in this example isthe |eiπ/4

state, represented in the standard basis as |0

+eiπ/4|1

. The quantum circuit uses fourteen qubits 32-45 storing a set ofhigh-fidelity initial states, including five copies of the plus state,stored in physical qubits 34-36, 41, and 45 representing the logicalqubits, and nine copies of a zero state, as well as fourteen copies ofthe |eiπ/4

state. It will be appreciated that each of the high-fidelity states canbe generated either via initialization of a qubit to the ground (zero)state in the standard basis or via a Hadamard gate operation on such aninitialized qubit to provide the plus state.

Each of the set of high-fidelity initial states is subjected to anencoding component 46 to encode the fourteen physical qubits representedin the set of initial states with a CCS quantum code having the propertythat an encoded −π/4 Z-rotation on each of the logical qubits can beimplemented via a transversal π/4 Z-rotation across all of the qubits.In general, the CCS quantum code is generated by puncturing a classicalcode having desired properties and mapping the generators of thecodewords within the punctured code to stabilizer generators and logicaloperators in a quantum code. In this instance, a first set ofunpunctured generators can represent X-type stabilizers and a second setof punctured generators can represent X-type logical Pauli operators. Inthe illustrated implementation, the encoding is performed via a seriesof controlled NOT (CNOT) gates, although it will be appreciated that theencoding can be performed via any appropriate set of Clifford gates.

In the illustrated implementation, the classical code is selected suchthat all of the codewords have weight 0 (mod 8). The punctures arearranged such that no generator loses more than a single bit with avalue of one and the unpunctured generators span the remaining bits.Distillation can be performed taking the unpunctured generators to be(under a mapping of zero to the identity matrix, I, and one to the PauliX operator, X) X-type stabilizers and the punctured generators to beX-type logical Pauli operators. It will be appreciated that any othermapping of the codeword to stabilizer generators and logical operatorsthat is equivalent within single qubit Clifford operations can also beutilized. In the example of FIG. 2, the classical code used is a firstorder Reed-Muller code, with a rate of five and a length of sixteen. Agenerator matrix for the Reed-Muller code can be represented as:

$\begin{matrix}\begin{bmatrix}1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{bmatrix} & {{Eq}.\mspace{14mu} 1}\end{matrix}$

In the illustrated implementation, the code is doubly punctured, leavinga generator matrix for the doubly punctured code that can be representedas:

$\begin{matrix}\begin{bmatrix}0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{bmatrix} & {{Eq}.\mspace{14mu} 2}\end{matrix}$

It will be appreciated that, in the punctured code, the first twocodewords have had their weights reduced to weight 7 mod 8, while theweights of the last three codewords are unaffected. Accordingly, thestabilizer generators for the quantum circuit can be represented as:

$\begin{matrix}\begin{bmatrix}X & X & I & I & X & X & I & I & X & X & I & I & X & X \\I & I & X & X & X & X & I & I & I & I & X & X & X & X \\I & I & I & I & I & I & X & X & X & X & X & X & X & X\end{bmatrix} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

The X-type logical Pauli operators can be represented as:X ₁=IXXIIXXIIXXIIXX ₂=IXIXIXIXIXIXIX  Eq. 4

The effect of the encoding component 46 is to place the qubits storingthe high-fidelity initial states into a state comprising a superpositionassociated with the stabilizer generators and the X-type logical Paulioperators. In accordance with an aspect of the present invention, arotation component 48 consumes the fourteen prepared low fidelity |eiπ/4

states to apply a rotation transversally across the plurality of qubits.In the circuit of FIG. 2, the rotation is a π/4 Z-rotation.

It will be appreciated that the π/4 Z rotation of FIG. 2 is simply aspecial case of a more general class of π/g Z rotations. FIG. 3illustrates one circuit 60 that might be used in the rotation component48 of FIG. 2 to perform a π/g Z rotation using a prepared |eiπ/g

ancilla state. The circuit 60 includes a target qubit 62 and an ancillaqubit 64 storing the |eiπ/g

ancilla state. A controlled not (CNOT) gate 66 is performed targetingqubit 62 and controlled by the ancilla qubit 64. The target qubit 62 isthen measured at a measurement assembly 68 that measures the targetqubit in the standard basis. A rotation component 70 conditionallyapplies a Z-rotation of 2π/g to the ancilla qubit 64 based upon themeasurement of the target qubit. Specifically, the 2π/g Z-rotation,followed by an X gate, is applied only if the target qubit is measuredto be in the excited (i.e., “1”) state. It will be appreciated that forsome values of g (e.g., multiples of four), the rotation component willbe a Clifford gate. For others, addition resource states may benecessary to implement the rotation using Clifford gates. After theconditional rotation, the ancilla qubit 64 will be in a staterepresenting the state of the target qubit 62 with a π/g rotationapplied.

Returning to FIG. 2, the rotated states are then provided to a decodingcomponent 50 that retrieves the logical qubits from the physical qubits32-45 comprising the quantum code. In the illustrated implementation,the decoding component 50 utilizes a set of CNOT gates similar to thatof the encoding component 46, but implemented in the reverse order.After decoding, each of the physical qubits corresponding to Z-typestabilizers (which in turn correspond to the physical qubits 32, 33,37-40, and 42-44 that began in the ground state) are measured in thestandard (Z) basis. Each of the physical qubits corresponding to X-typestabilizers (which correspond to the physical qubits 36, 41, and 45) aremeasured in the sign (X) basis. If any of these measured values are notthe expected value, it can be assumed that an error occurred in thedistillation process, and the states stored in the remaining physicalqubits 34 and 35 can be discarded. If all of the measurements providethe expected values, the states stored in the remaining physical qubits34 and 35 will be resource states having an increased fidelity ascompared to the resource states consumed at the rotation element 48.

The inventor has determined that the systems and methods of the presentinvention are generalizable for distillation routines for |eiπ/g

by using a classical code whose codewords all have weight 0 mod 2g.Direct distillation of |eiπ/g

states for use in π/g Z-rotations may prove to be much more resourceefficient than generating π/g Z-rotations using π/4 Z-rotations andHadamard gates.

The number of input copies of the |eiπ/g

state per each output copy of the |e−iπ/g

state may be regarded as a measure of the efficiency of a resource statedistillation routine. By this metric, referred to herein as stateoverhead, the best existing magic-state distillation routine to achievea quadratic reduction in the output error probability at the time of thepresent invention had an overhead of five input states per output state.The example distillation circuits shown in FIGS. 2 and 4 have anoverhead of seven input states per output state, but the same techniquecan be used to approach arbitrarily close to an overhead of three inputstates per output state.

As general implementation of transversal rotation technique of FIGS. 2and 4 can be performed by starting with a classical code formed byconcatenating an m-bit parity code, where m is a multiple of four, witha four-bit repetition code, and then augmenting the code by theindependent generators of the m-bit repetition code concatenated with afour-bit parity code. In other words, the method begins with a classicalcode whose (m+1)×4m generator matrix has the form:

$\begin{matrix}\begin{bmatrix}P & P & P & P \\0 & R & 0 & R \\0 & 0 & R & R\end{bmatrix} & {{Eq}.\mspace{14mu} 5}\end{matrix}$

where P is a matrix of the generators of an m-bit parity code, R is avector of m ones, 0 is a vector of m zeros, and m is a multiple of four.

The code is punctured at the first m−2 bits and the punctured generatorsare used to define logical X operators and the unpunctured ones todefine X-type stabilizers. Transversal rotation distillation using sucha quantum code yields a quadratic reduction in the error probability atthe cost of (4m−(m−2))/(m−2)=(3m+2)/(m−2) input states per output state,which approaches three in the limit of large m. It will be appreciated,however, that the coefficient of the marginal output error rate willincrease with increasing m.

Transversal rotation distillation routines can also be derived fromclassical codes having generators of the form:

$\begin{matrix}\begin{bmatrix}D & D \\0 & R\end{bmatrix} & {{Eq}.\mspace{14mu} 6}\end{matrix}$

where D is a matrix of the generators of a doubly-even code, R is avector of k ones, 0 is a vector of k zeros, and k is a multiple ofeight.

For example, taking D to be the extended binary Golay code andpuncturing ten coordinates gives a transversal rotation distillationroutine with quadratic error suppression at a cost of 3.8 (38-to-10)input states per output state. In this specific case, the constructionof Eq. 5 gives a superior 38-to-10 routine.

Transversal rotation distillation routines with higher orders of errorsuppression can also be utilized. One method to achieve stronger errorsuppression is to puncture only the first term of a 4m-bit classicalcode, where m is a multiple of four, of the form shown in Eq. 5. When mis equal to four, the corresponding transversal rotation distillationyields a cubic reduction in the error probability at the cost of fifteeninput states per output state. Stronger error suppression can beachieved by starting from more complex classical codes, such as thehigher order Reed-Muller codes. For example, puncturing the 128-bitsecond-order Reed-Muller code, RM(2; 7), at bits 1, 2, 3, 5, 9, 17, 33,and 65 (for a standard ordering of bits) gives a 120-to-8 distillationroutine with quartic error suppression.

FIG. 4 illustrates another example of a system 150 for generating atleast one high fidelity magic state for the set of Clifford gates from aplurality of relatively low fidelity magic states in accordance with anaspect of the present invention. While the number of input magic statesexpended per output magic state is important, it is not the onlyquantity relevant to the efficiency of a distillation routine. If weenvision magic state distillation as performed using qubits and Cliffordgates that have been made highly robust through substantial effort, thenthe number of locations in a distillation routine affects the error ratethat must be achieved in these components such that no error is likelyto be observed. Thus, another important quantity is the number oflocations in a distillation routine per output state, referred to hereinas location overhead. State and location overhead can recommenddifferent routines, particularly when only a round or two ofresource-state distillation is required, which is the expected situationfor some quantum computing architectures. Consequently, the minimizationof location overhead warrants independent consideration.

In the implementation of FIG. 3, it will be noted that the encoder needonly prepare the |+> state on each of the logical qubits. For CSS codes,the inventor has determined that encoded state can be prepared bypreparing |+> transversally on each of the physical qubits, measuringeach of the Z-stabilizers, and then applying correcting X gates asindicated by the measurement outcomes. This encoder is not necessarilymore compact than direct encoding, but the measurement and correctioncan be pushed through to the right side of the transversal physical π/4Z-rotations. The Z-stabilizer measurements are unchanged by thistransformation, but the correcting X gates are transformed intocorrecting Z(π/2)X gates. The initial |+> preparations with the π/4Z-rotations can be combined into |eiπ/4

states, and a sequence of circuit identities can be used to push themeasurement and correction of the code subspace can be toward the end ofthe circuit until it directly proceeds the final measurements in thecircuit.

At that point, the Z stabilizer measurements are simply measurements onindividual qubits, but the correcting unitaries have become acomplicated sequence of Z(π/2) rotations, controlled Z gates, and Xgates. Controlled Z gates acting on a qubit measured in the Z eigenbasiscan be replaced by Z gates controlled on the qubit's classicalmeasurement outcome. Z(π/2) rotation gates acting on a qubit measured inthe Z eigenbasis have no effect and X gates trivially flip the value ofthe next Z measurement. Consequently, these gates and the final Zmeasurements can be omitted. In addition, X gates have no effect on theoutcome of measurements in the X eigenbasis, so X gates immediatelypreceding these measurements can also be dropped.

This completes the circuit's transformation to acompacted-transversal-preparation (CTP) distillation circuit,represented by the system 150 of FIG. 4. Specifically, the illustratedsystem begins with a plurality of physical qubits 152-155 each storing arelatively low fidelity state from a family of states that areequivalent to |eiπ/4

By “equivalent,” it is meant that each state in the family of states canbe derived from the |eiπ/4

state via one-qubit Clifford unitary operators.

The system 150 includes a decoder 160 that applies a decoding process tothe low fidelity states stored in the plurality of physical qubits152-155 according to a selected quantum code. In accordance with anaspect of the present invention, the quantum code is derived from apunctured classical code having a first set of code words with a weightof zero mod eight and a second set of code words with a weight of sevenmod eight. It will be appreciated that by “derived from”, it is meantthat the generators for the quantum code were generated via a standardmapping process (i.e., each 0 bit to an identity operator and each onebit to an X operator) or a Clifford-equivalent mapping process, in whichthe operators representing each bit can be derived via one-qubitClifford unitary operators applied to the standard mapping.

The decoder 160 includes a plurality of quantum gates that decode theZ-type stabilizer generators into individual physical qubits. Thedecoder 160 further includes a plurality of measurement assemblies formeasuring the physical qubits corresponding (after decoding) to Z-typestabilizer generators. These measurements can be recorded as a first setof measurement values, for example, on a non-transitory computerreadable medium at an associated classical computer. The remainingphysical qubits store the X-type stabilizer generators and the logicalqubits in a manner determined by the outcomes obtained from measurementof the Z-type stabilizer generators.

A correction component 170 includes a plurality of quantum gates, eachconfigured to conditionally apply an associated operation on the propersubset of physical qubits that remain unmeasured according to the firstset of measurement values. By “conditionally apply”, it is meant thateach rotation gate is applied to associated qubit only if one or more ofthe first set of measurement values assumes a given value. It will beappreciated that the gates within the correction component can beclassically controlled via an associated computer (not shown) andvarious classical control mechanisms. The effect of the conditionalgates is to put a first set of the physical qubits corresponding to thelogical qubits in a state that is expected to be, absent error, thedesired magic state. A plurality of measurement assemblies are providedto measure a set of the remaining physical qubits, which represent theX-type stabilizers in the quantum code, to provide a second set ofmeasurement values. From the second set of measurements, it can bedetermined if the system 150 has experienced one of various types oferrors. Accordingly, if the second set of measurement values assumecorresponding desired values, the remaining qubits are retained as highfidelity magic states. If one or more of the second set of measurementvalues do not assume the correct value, the remaining qubits can becorrected or discarded.

FIG. 5 illustrates one implementation of the system of FIG. 4 as aquantum circuit 200 for performing magic state distillation inaccordance with an aspect of the present invention. The illustratedquantum circuit 200 includes a plurality of physical qubits 202-215 eachprepared in a relatively low fidelity magic state from a family ofstates equivalent to the +1 Hadamard eigenstate. For brevity, this stateis referred herein as the |eiπ/4

state. A decoding component 220 is applied to decode the Z-typestabilizer generators to individual physical qubits. For the specificapplication of magic state distillation, the quantum code is based on apunctured classical code selected such that all of the codewords haveweight 0 (mod 8), with the punctures are arranged such that no generatorloses more than a single bit with a value of one and the unpuncturedgenerators span the remaining bits. In the illustrated decoder 220, afirst order, sixteen-bit Reed-Muller code is utilized. As shown in thediagram, the decoder 220 utilizes a series of two qubit gates, in thiscase, controlled CNOT gates, to decode the prepared states.

The decoder 220 includes a plurality of measurement assemblies and aspart of the decoding process, a first set of physical qubits 202, 203,207-210, 212-214 are measured in the standard (Z) basis. Thesemeasurements are recorded as a first set of measurements. The first setof measurements and the remaining qubit states are provided to acorrection assembly 230. The correction assembly 230 includes aplurality of quantum gates that are classically controlled according tothe first set of measurements to conditionally apply quantum operationsto a second set of qubits 204-206, 211, and 215. The plurality ofquantum gates can include any appropriate Clifford gate, and in theillustrated example, a series of controlled Z gates are used as well asX gates, and variable Z rotations.

Along with determining whether a given one of the plurality of quantumgates will be utilized, the first set of measurement values are used todetermine a specific angle for each Z rotation, as well as whether eachof the physical qubits 206, 211, and 215 representing X-type stabilizersof the quantum code will be measured in the sign (X) basis or the Ybasis. Regardless of which basis is used, a second set of measurementvalues derived from these measurements can be compared to an expectedset of values. If any of these measured values are not found to be anexpected value, it can be assumed that an error occurred in thedistillation process, and the states stored in the remaining(unmeasured) physical qubits 204 and 205 can be discarded. If all of themeasurements provide the expected values, the states stored in theremaining physical qubits 204 and 205 will be magic states having anincreased fidelity as compared to the initial states.

In view of the foregoing structural and functional features describedabove in FIGS. 1-5, example methodologies will be better appreciatedwith reference to FIGS. 6 and 7. While, for purposes of simplicity ofexplanation, the methodologies of FIGS. 6 and 7 are shown and describedas executing serially, it is to be understood and appreciated that thepresent invention is not limited by the illustrated order, as someactions could in other examples occur in different orders and/orconcurrently from that shown and described herein.

FIG. 6 illustrates a method 250 for generating at least one highfidelity resource state. At 252, a classical code is selected in whichall of the codewords associated with the code have a weight of zero mod2g, where g is a positive integer power of two. In one implementation,this is a first order, sixteen-bit Reed-Muller code. At 254, theclassical code is punctured such that a first set of codewords retain aweight of zero mod 2g and a second set of codewords has a weight of 2g−1mod 2g. In one implementation, the code is doubly punctured, with thefirst two bits of each codeword being removed. At 256, the first set ofgenerators (for the classical code) are mapped to a set of stabilizergenerators. At 258, the second set of generators to a set of logicaloperators. In one implementation, binary zeros within the codewords aremapped to an identity operator, and binary ones within the codewords aremapped to Pauli X operators to provide the generators and operators.

At 260, a set of relatively low fidelity resource states are prepared ina plurality of physical qubits. In one implementation, high fidelityinitial states are encoded using a quantum code represented by the setof stabilizer generators and the set of logical operators, and arotation is applied transversally across the physical qubits. In anotherimplementation, each physical qubit is simply prepared in the desiredresource state. At 262, a decoding process is performed on the set ofrelatively low fidelity resource states according to the quantum coderepresented by the set of stabilizer generators and the set of logicaloperators. This begins the process of extracting a set of logical qubitsfrom the quantum code. At 264, measurement of the physical qubitscorresponding to Z-type stabilizers is performed and at 266 this set ofmeasurement values is used to control selective corrections to theremaining physical qubits. Following any necessary corrections, theunmeasured physical qubits can be divided into a first set correspondingto the logical qubits and a second set corresponding to the X-typestabilizers. At 268, each of the physical qubits in the second set aremeasured to provide a second set of measurement values. The second setof measurement values enable the detection of errors introduced by thelow fidelity states. If all of the second set of measurements assumedesired values, the remaining qubits will be expected to contain highfidelity representations of the resource state.

FIG. 7 illustrates a method 300 for generating at least one highfidelity resource state. At 302, a classical code is selected in whichall of the codewords associated with the code have a weight of zero mod2g, where g is a positive integer power of two. In one implementation,this is a first order, sixteen-bit Reed-Muller code. At 304, theclassical code is punctured such that a first set of codewords retain aweight of zero mod 2g and a second set of codewords has a weight of 2g−1mod 2g. In one implementation, the code is doubly punctured, with thefirst two bits of each codeword being removed. At 306, the first set ofgenerators (for the classical code) are mapped to a set of stabilizergenerators. At 308, the second set of generators to a set of logicaloperators. In one implementation, binary zeros within the codewords aremapped to an identity operator, and binary ones within the codewords aremapped to Pauli X operators to provide the generators and operators.

At 310, a set of physical qubits are prepared in a plurality of initialstates. In one implementation, these states can include a plurality ofplus states and a plurality of zero states. At 312, the physical qubitsare encoded using a quantum code represented by the set of stabilizergenerators and the set of logical operators. At 314, a rotation isapplied transversally across the physical qubits. At 316, a decodingprocess is performed on the qubits according to the quantum coderepresented by the set of stabilizer generators and the set of logicaloperators. At 318, measurement of the physical qubits corresponding toZ-type stabilizers and X-type stabilizers is performed. If all of themeasurements assume desired values, the remaining physical qubits willbe expected to store high fidelity representations of the resourcestate.

What have been described above are examples of the present invention. Itis, of course, not possible to describe every conceivable combination ofcomponents or methodologies for purposes of describing the presentinvention, but one of ordinary skill in the art will recognize that manyfurther combinations and permutations of the present invention arepossible. Accordingly, the present invention is intended to embrace allsuch alterations, modifications, and variations that fall within thescope of the appended claims.

What is claimed is:
 1. A system for generating at least one highfidelity resource state from a plurality of relatively low fidelityresource states, the system comprising: an encoding circuit that encodesa set of logical qubits into a plurality of physical qubits according toa punctured classical code; a rotation component that consumes therelatively low fidelity resource states to apply a rotation to a statestored in each physical qubit around an axis of the Bloch sphere of π/g,where g is a positive integer power of two; a decoding circuit thatdecodes the rotated logical states stored in the physical qubits toprovide the set of logical qubits stored in a proper subset of theplurality of physical qubits, at least one of the set of logical qubitscomprising a high fidelity magic state.
 2. The system of claim 1,wherein the encoding circuit comprises a series of two qubit gatesconfigured to place the plurality of physical qubits into a statecomprising a superposition associated with a set of stabilizergenerators and X-type logical Pauli operators associated with thepunctured classical code.
 3. The system of claim 2, wherein the decodingcircuit comprises the series of two qubit gates performed in a reverseorder from that of the encoding circuit.
 4. The system of claim 2, inwhich the series of two qubit gates comprises a series of controlled NOT(CNOT) gates.
 5. The system of claim 1, wherein the rotation component,for each physical qubit, comprises: a two qubit gate configured toentangle the state stored in the physical qubit with the relatively lowfidelity resource state consumed by the rotation, the relatively lowfidelity resource state being stored in an ancilla qubit; a measurementassembly configured to measure the physical qubit to provide ameasurement value; and a conditional rotation gate configured to apply arotation to the state stored in the ancilla qubit around the axis of theBloch sphere of 2π/g if the measurement value assumes a desired value,such that the state stored in the ancilla qubit represents the statestored in the physical qubit rotated around the axis of the Bloch sphereby π/g.
 6. A system for generating at least one high fidelity magicstate for a set of Clifford gates, the system comprising: a plurality ofphysical qubits each storing a relatively low fidelity state equivalentto |eiπ/4

; a decoder comprising: a first plurality of quantum gates that decodethe states stored in the physical qubits to provide a first strictsubset of the plurality of physical qubits corresponding to Z-typestabilizer generators according to a quantum code derived from apunctured classical code having a first set of codewords with a weightof zero mod 8 and a second set of codewords with a weight of 7 mod 8;and a first plurality of measurement assemblies for measuring the firstsubset of the plurality of physical qubits that correspond to Z-typestabilizer generators to provide a first set of measurement values; anda correction component comprising: a second plurality of quantum gates,each configured to conditionally apply an associated operation to astate stored in at least one physical qubit that is not part of thefirst subset according to the first set of measurement values; and asecond set of measurement assemblies for measuring a second strictsubset of the plurality of physical qubits that, after correction,correspond to X-type stabilizer generators.
 7. The system of claim 6,wherein the first set of measurement assemblies are configured tomeasure the first subset of the plurality of physical qubits thatcorrespond to Z-type stabilizer generators in the standard basis and thesecond set of measurement assemblies are configured to measure thesecond subset of physical qubits that correspond to X-type stabilizergenerator in one of the sign basis and the Y-basis.
 8. The system ofclaim 7, wherein the one of the sign basis and the Y basis is selectedaccording to the first set of measurement values.